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Numerical solution of the exact equations for capillary–gravity waves

Published online by Cambridge University Press:  19 April 2006

Leonard W. Schwartz  and
Jean-Marc Vanden-Broeck
Leonard W. Schwartz
Affiliation:
Department of Applied Mathematics, University of Adelaide, S. Australia
Jean-Marc Vanden-Broeck
Affiliation:
Department of Applied Mathematics, University of Adelaide, S. Australia Present address: Courant Institute of Mathematical Sciences, New York University, New York.
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Abstract

A numerical method is presented for the computation of two-dimensional periodic progressive surface waves propagating under the combined influence of gravity and surface tension. The dynamic boundary equation is used in its exact nonlinear form. The procedure involves a boundary-integral formulation coupled with a Newtonian iteration. Solutions of high accuracy can be achieved over much of the range of wavelengths and heights including limiting waves. A number of different continuous families of solutions have been produced, all of which ultimately exhibit closed bubbles at their troughs. The so-called critical wavelengths are less important than have been previously assumed; the number of possible wave forms does increase with increasing wavelength, however.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 95 , Issue 1 , 14 November 1979 , pp. 119 - 139
Copyright
© 1979 Cambridge University Press

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